Let \(\gamma =0\) . Then () holds if \({N^{1/\beta } \hat{f}(U_F(N)) (\log (T))^{1/\beta }}/{T^{p}}\rightarrow 0\) . By lemma 1.2.9 in [1]} \({\hat{f}(U_F(N)) }/{U_F(N)} \rightarrow 0\)  and \(U_F(N)\in RV_0\) . Thus
\(\frac{N^{1/\beta } \hat{f}(U_F(N)) (\log (T))^{1/\beta }}{T^{p}} & = \frac{N^{1/\beta } (\log (T))^{1/\beta }}{T^{p}} \dfrac{\hat{f}(U_F(N))}{U_F(N)} U_F(N) \dfrac{N^{-\gamma ^{\prime }}}{N^{-\gamma ^{\prime }}}\\& = \dfrac{ N^{-1/\beta -\gamma ^{\prime }}(\log (T))^{1/\beta } }{T^{p}} \dfrac{\hat{f}(U_F(N))}{U_F(N)} \dfrac{U_F(N)}{N^{-\gamma ^{\prime }}} \rightarrow 0.\) 
